Analysis of Brownian motion and particle size effects on the thermal behavior and cooling performance of microchannel heat sinks

Abstract

In the present paper the thermal performance of nanofluid flow in microchannel heat sinks (MCHSs) was analyzed using three different nanofluids. The fluid flow and heat transfer in the MCHS are modeled using the Darcy–Brinkman–Forchheimer and two-equation model, respectively. In order to check the validity of the results, they were compared with available experimental and numerical data in the literature and excellent agreement was found between them. The key novelties of present work are (i) using a novel and complex temperature dependent thermal conductivity model for nanofluids, based on Brownian motion induced micro mixing, (ii) using new nanofluids with three types of nanoparticle materials i.e. aluminum oxide (Al₂O₃), zinc oxide (ZnO) and copper oxide (CuO) and 60:40 (by mass) EG–water for the base fluid, (iii) investigating the effect of nanoparticles size and Brownian motion of particles on thermal performance of MCHS.

Introduction

Advances in microfabrication have led to development of different kinds of microdevices such as micro-pumps, micro-turbines, micro-mixers, micro-motors, micro-sensors, micro-reactors, micro-heat exchangers, etc. Microchannel heat sinks due to their high thermal performance are one of the most attractive devices for cooling electronic components. Some of their distinct features with respect to conventional heat dissipating devices are their small size and volume per heat load, high heat transfer performance and small coolant requirements. Thermal performance of these devices is function of their geometry, i.e., dimensions and number of channels, thermal properties of coolant and velocity of flow in microchannels. Several investigators have explored fluid flow and heat transfer characteristics in MCHSs. For example, Kim and Kim [1] analytically studied thermal performance of MCHSs using two-equation model for heat transfer and a Brinkman extended Darcy model for fluid flow. Their results agreed well with experimental measurement and those obtained by fin theory model. Chen and Ding [2] modeled forced convection heat transfer in MCHSs using Forchheimer–Brinkman-extended Darcy model to account for both boundary effects and fluid inertia.

In the recent years, nanofluids (NFs) have been proved to have great potential for enhancing thermal conductivity and consequently heat transfer coefficient of heat transfer devices [3], [4]. Therefore, using nanofluids as working fluid is well suited for use in high performance compact heat exchangers and heat sinks used in electronic equipments. The enhancement of thermal conductivity NFs depends on particle diameter and volume fraction, thermal conductivities of base fluid and nanoparticles as well as Brownian motion of nanoparticles that is a key mechanism in thermal conductivity enhancement. Over the past years, various models have been proposed for calculating thermal conductivity of nanofluids. Concise review of these models is provided in Ref [3], [4]. Due to unusual heat transfer characteristics of nanofluids, effective thermal conductivity of nanofluids cannot be explained by conventional model for mixtures because they originated from continuum formulations which assume diffusive heat transfer in both solid and fluid phases and involve only volume fraction and the particle shape while Brownian motion at nanoscale and molecular levels is one of the main mechanisms which should be taken into account in nanofluids thermal conductivity models. Wang et al. [5] argued that the thermal conductivities of nanofluids should be dependent on particle structure and the Brownian motion and inter-particle forces. Koo and Kleinstreuer [6] discussed the effects of thermo-phoretic, Brownian and osmo-phoretic motions on effective thermal conductivity of nanofluids. They found that the roles of thermo-phoretic and osmo-phoretic motions are much less important than the Brownian motion. Some of novel and efficient models for thermal conductivity of nanofluids, which take into the effect of Brownian motion, are summarized as follows:

Xuan et al. [7] presented a model considering the Brownian motion of nanoparticles and their aggregation.

$$k_{\text{nf}}=\frac{k_p+2k_{\text{bf}}-2\left(k_{\text{bf}}-k_p\right)\varphi }{k_p+2k_{\text{bf}}+\left(k_{\text{bf}}-k_p\right)\varphi }{k}_{\text{bf}}+\frac{\varphi {\rho }_p{C}_{p,p}}{2}\times \sqrt{\frac{K_{b}T}{3\mathrm{\pi }{\mu }_{\text{bf}}{r}_c}}$$

Chon et al. [8] developed an empirical correlation for the thermal conductivity of Al₂O₃ as a function of temperature, nanoparticles volume fraction and size for a wide range of temperature between 21 °C and 70 °C.

$$\frac{k_{\text{nf}}}{k_{\text{bf}}}=1+64.7{\varphi }^{0.7460}{\left(\frac{d_{\text{bf}}}{d_p}\right)}^{0.3690}{\left(\frac{k_p}{k_{\text{bf}}}\right)}^{0.7446}\mathit{P}{\mathit{r}}^{0.9955}\mathit{R}{\mathit{e}}^{1.2321}$$

Prasher et al. [9] proposed a convective-conductive model called the multi-sphere Brownian model (MSBM) which considers the effect of Brownian motion and the influence of interfacial thermal resistance between different fluids and nanoparticles.

$$\frac{k_{\text{nf}}}{k_{\text{bf}}}=\left(1+A\mathit{R}{\mathit{e}}_{\mathit{b}}^{\mathit{m}}\mathit{P}{\mathit{r}}^{0.333}\varphi \right)\left(\frac{\left[k_p\left(1+2\overline{\alpha }\right)+2k_m\right]+2\varphi \left[k_p\left(1-\overline{\alpha }\right)-k_m\right]}{\left[k_p\left(1+2\overline{\alpha }\right)+2k_m\right]-\varphi \left[k_p\left(1-\overline{\alpha }\right)-k_m\right]}\right)$$

This model has a very good agreement with some of recent Al₂O₃ experimental data set [10]. However (i) it cannot predict the thermal conductivity enhancement trend for the experimental results of Li and Peterson [11], (ii) Comparisons show that this model is not suitable when the particles size are too large (≥125 nm) or too small (≤11 nm), (iii) It is not in good agreement with the experimental data of Chon et al. [8].

Jang and Choi [12] developed a theoretical model based on Kapitza resistance, kinetic theory, and Brownian motion. Their model involves four modes contributing to the energy transport in nanofluids. The correlation is given as

$$k_{\text{nf}}=k_{\text{bf}}\left(1-\alpha \right)+{\beta }_1{k}_p\varphi +C_1\frac{d_{\text{bf}}}{d_p}{k}_{\text{bf}}\mathit{R}{\mathit{e}}_{d_p}^2\mathit{Pr}\varphi$$

Kleinstreuer and Li [13] in a comparison study showed that when the fluid temperature changes, the effective thermal conductivity theory of Jang and Choi [12] cannot consistently match measured thermal performances of nanofluids.

Koo and Kleinstreuer [14], [15] proposed a thermal conductivity model that takes into account the effect of temperature, particle volumetric concentration, particle size and properties of base fluid as well as nanoparticles subjected to Brownian motion. The range of temperature and volume fraction for this model is between 293 and 325 K and 1% < φ < 4%, respectively. The model can be written as

$$k_{\text{nf}}=\frac{k_p+2k_{\text{bf}}-2\left(k_{\text{bf}}-k_p\right)\varphi }{k_p+2k_{\text{bf}}+\left(k_{\text{bf}}-k_p\right)\varphi }{k}_{\text{bf}}+5\times {10}^4\varphi \beta {\rho }_{\text{bf}}{C}_{p,\text{bf}}\times \sqrt{\frac{K_{b}T}{{\rho }_p{d}_p}}f(T,\varphi ,\text{etc}\text{.})$$

The function f and $\beta$, must be determined semi-empirically.

Recently, Vajjha and Das [16] based on their experimental data extended the Koo and Kleinstreuer [14], [15] model for 60:40 (by mass) Ethylene Glycol–Water (EG/W) as base fluid. The applicable range of particle size and temperature for their model is between 29 and 77 nm and 293 − 325 K. Comparisons between different models show that the model proposed by Koo and Kleinstreuer [14], [15] predicts thermal conductivity of nanofluids better than other available models [10], [16]. Therefore, in the present paper we used the extended form of this model, i.e, the model developed by Vajjha and Das [16], for the study of thermal performance of nanofluid-cooled MCHSs.

According to literature, there are many works related to thermal performance of MCHSs with water based nanofluids but to the knowledge of authors there are no theoretical investigation for other base fluids such as 60% ethylene glycol and 40% water by mass (60:40 EG/W) which is most widely used fluid for cold regions of the world. Furthermore, there is no work to study the effect of particle size and Brownian motion on thermal behavior of nanofluid-cooled MCHSs with 60:40 EG/W base fluid. For this reason, the main scope of this paper is to investigate the thermal performance of nanofluid-cooled MCHS with three types of nanoparticle material i.e. aluminum oxide (Al₂O₃), zinc oxide (ZnO) and copper oxide (CuO) dispersed in 60:40 EG/W as base fluid. In order to calculating thermal conductivity of nanofluid a novel, complex and efficient temperature dependent thermal conductivity model that take into account the effect of particle size and Brownian motion on nanofluid thermal conductivity is used for the first time in theoretical study of MCHSs. Furthermore, micro channel heat sink is modeled as a "fluid-saturated porous medium" with a nonlinear and more complicated model for fluid flow i.e. Forchheimer–Brinkman-extended Darcy equation. The results are compared to the available experimental data and a good match was found between them when the effect of inertial force is taken into account. Finally, the effect of type of nanoparticles, particle diameter, volume fraction, and Brownian motion are examined on performance of MCHSs.